Fock Space Inventory: Spaces, States, and Operators After Second Quantization

This page is a reference for what is what in a quantum field theory once second quantization has been performed. It expands on QFT/preliminaries.md § Fock Space and § States vs. Fields, and is meant to clarify the distinct roles of state vectors, field operators, ladder operators, and c-number mode coefficients.

The conventions and notation follow the historical-route QED notes, but the content is general.

0.0 Mathematical Preliminaries for Fock Space

The Fock-space construction below uses three pieces of linear-algebraic machinery: the tensor product, the direct sum, and the (anti)symmetric tensor power. The basic versions of $\otimes$ and $\oplus$ are recapped in QM/preliminaries.md § Notation Key; this section adds the parts specific to identical-particle Hilbert spaces.

0.0.1 Tensor Powers

For a Hilbert space $\mathcal{H}$, the $N$-fold tensor power is

$${\mathcal{H}}^{\otimes N}\equiv \underset{N\text{ factors}}{\underbrace{\mathcal{H}\otimes \mathcal{H}\otimes \cdots \otimes \mathcal{H}}}.$$

Vectors are linear combinations of product states $|{\varphi }_1⟩\otimes |{\varphi }_2⟩\otimes \cdots \otimes |{\varphi }_N⟩$ with $|{\varphi }_i⟩\in \mathcal{H}$. The inner product is defined factor-by-factor and extended by linearity:

$$⟨{\varphi }_1\otimes \cdots \otimes {\varphi }_N|{\chi }_1\otimes \cdots \otimes {\chi }_N⟩=\prod _{i=1}^N⟨{\varphi }_i|{\chi }_i⟩.$$

This is the natural Hilbert space for $N$ distinguishable particles. For identical particles, the symmetry of the wavefunction under particle exchange must be specified — leading to the symmetric / antisymmetric subspaces below.

0.0.2 The Symmetric Group $S_N$

A permutation of $\{1,2,\dots ,N\}$ is a bijection $\sigma :\{1,\dots ,N\}\to \{1,\dots ,N\}$. The set of all such permutations forms the symmetric group $S_N$, with $|S_N|=N!$.

Each permutation has a sign $\mathrm{sgn}(\sigma )=\pm 1$:

• $\mathrm{sgn}(\sigma )=+1$ if $\sigma$ is a product of an even number of transpositions (swaps).
• $\mathrm{sgn}(\sigma )=-1$ if $\sigma$ is a product of an odd number of transpositions.

For example, in $S_3$: the identity has sign $+1$; any single swap (e.g. $1\leftrightarrow 2$) has sign $-1$; the cyclic permutation $1\to 2\to 3\to 1$ has sign $+1$ (it's a product of two transpositions).

0.0.3 (Anti)Symmetrization Projectors

Permutations act on ${\mathcal{H}}^{\otimes N}$ by permuting factors:

$${\hat{\Pi }}_{\sigma }|{\varphi }_1⟩\otimes \cdots \otimes |{\varphi }_N⟩=|{\varphi }_{{\sigma }^{-1}(1)}⟩\otimes \cdots \otimes |{\varphi }_{{\sigma }^{-1}(N)}⟩.$$

Two orthogonal projectors are constructed by averaging over $S_N$:

$$\hat{S}=\frac{1}{N!}\sum _{\sigma \in S_N}{\hat{\Pi }}_{\sigma }\text{(symmetrizer)},$$ $$\hat{A}=\frac{1}{N!}\sum _{\sigma \in S_N}\mathrm{sgn}(\sigma ){\hat{\Pi }}_{\sigma }\text{(antisymmetrizer)}.$$

Both satisfy ${\hat{S}}^2=\hat{S}$, ${\hat{A}}^2=\hat{A}$, ${\hat{S}}^{†}=\hat{S}$, ${\hat{A}}^{†}=\hat{A}$.

0.0.4 Symmetric Tensor Power ${\mathrm{Sym}}^N(\mathcal{H})$

The $N$-fold symmetric tensor power is the image of $\hat{S}$:

$$\boxed{{\mathrm{Sym}}^N(\mathcal{H})\equiv \hat{S}{\mathcal{H}}^{\otimes N}=\{|\Psi ⟩\in {\mathcal{H}}^{\otimes N}:{\hat{\Pi }}_{\sigma }|\Psi ⟩=|\Psi ⟩\text{ for all }\sigma \in S_N\}.}$$

Vectors in ${\mathrm{Sym}}^N(\mathcal{H})$ are invariant under any permutation of the $N$ tensor factors. Explicitly, the symmetrized product of $N$ single-particle states is

$$|{\varphi }_1;{\varphi }_2;\cdots ;{\varphi }_N{⟩}_S\equiv \hat{S}(|{\varphi }_1⟩\otimes \cdots \otimes |{\varphi }_N⟩)=\frac{1}{N!}\sum _{\sigma \in S_N}|{\varphi }_{\sigma (1)}⟩\otimes \cdots \otimes |{\varphi }_{\sigma (N)}⟩.$$

For $N=2$:

$$|\varphi ;\chi {⟩}_S=\frac{1}{2}(|\varphi ⟩\otimes |\chi ⟩+|\chi ⟩\otimes |\varphi ⟩).$$

For $N=3$:

$$|\varphi ;\chi ;\xi {⟩}_S=\frac{1}{6}\sum _{\sigma \in S_3}|{\varphi }_{\sigma (1)}⟩\otimes |{\varphi }_{\sigma (2)}⟩\otimes |{\varphi }_{\sigma (3)}⟩.$$

This is the Hilbert space of $N$ identical bosons — the wavefunction is unchanged when any two particles are swapped, and the same single-particle state can be occupied any number of times.

Notation aliases used elsewhere: $S^N(\mathcal{H})$, ${\mathcal{H}}^{\odot N}$, ${⨂}_s^N\mathcal{H}$.

0.0.5 Antisymmetric (Exterior) Tensor Power ${\Lambda }^N(\mathcal{H})$

The $N$-fold antisymmetric (exterior) tensor power is the image of $\hat{A}$:

$$\boxed{{\Lambda }^N(\mathcal{H})\equiv \hat{A}{\mathcal{H}}^{\otimes N}=\{|\Psi ⟩\in {\mathcal{H}}^{\otimes N}:{\hat{\Pi }}_{\sigma }|\Psi ⟩=\mathrm{sgn}(\sigma )|\Psi ⟩\text{ for all }\sigma \in S_N\}.}$$

Vectors in ${\Lambda }^N(\mathcal{H})$ pick up a sign $\mathrm{sgn}(\sigma )$ under permutation. The antisymmetrized product of $N$ single-particle states is the Slater determinant

$$|{\varphi }_1;{\varphi }_2;\cdots ;{\varphi }_N{⟩}_A\equiv \hat{A}(|{\varphi }_1⟩\otimes \cdots \otimes |{\varphi }_N⟩)=\frac{1}{N!}\sum _{\sigma \in S_N}\mathrm{sgn}(\sigma )|{\varphi }_{\sigma (1)}⟩\otimes \cdots \otimes |{\varphi }_{\sigma (N)}⟩.$$

For $N=2$:

$$|\varphi ;\chi {⟩}_A=\frac{1}{2}(|\varphi ⟩\otimes |\chi ⟩-|\chi ⟩\otimes |\varphi ⟩).$$

In particular, $|\varphi ;\varphi {⟩}_A=0$: two identical fermions cannot occupy the same single-particle state. This is the Pauli exclusion principle, built directly into the algebra.

This is the Hilbert space of $N$ identical fermions.

Notation aliases used elsewhere: ${\bigwedge }^N\mathcal{H}$, ${\mathcal{H}}^{\wedge N}$, ${⨂}_a^N\mathcal{H}$.

0.0.6 Conventions for $N=0,1$

Both extreme cases give the same result for bosons and fermions:

• ${\mathrm{Sym}}^0(\mathcal{H})={\Lambda }^0(\mathcal{H})=\mathbb{C}$ — a one-dimensional space, identified with the vacuum sector ${\mathcal{H}}_0$ spanned by $|0⟩$.
• ${\mathrm{Sym}}^1(\mathcal{H})={\Lambda }^1(\mathcal{H})=\mathcal{H}$ — the single-particle sector is just $\mathcal{H}$ itself; there are no permutations to (anti)symmetrize.

Bosons and fermions therefore differ only for $N\ge 2$.

0.0.7 Hilbert Direct Sum

The Hilbert direct sum ${⨁}_{n=0}^{\infty }{\mathcal{H}}_n$ of countably many Hilbert spaces ${\mathcal{H}}_n$ consists of sequences $({\psi }_n{)}_{n=0}^{\infty }$ with ${\psi }_n\in {\mathcal{H}}_n$ and finite total norm:

$$\sum _{n=0}^{\infty }\Vert {\psi }_n{\Vert }_{{\mathcal{H}}_n}^2<\infty .$$

The inner product is

$$⟨({\varphi }_n)|({\psi }_n)⟩=\sum _{n=0}^{\infty }⟨{\varphi }_n|{\psi }_n{⟩}_{{\mathcal{H}}_n}.$$

Different summands are mutually orthogonal — a vector in ${\mathcal{H}}_n$ is orthogonal to a vector in ${\mathcal{H}}_m$ for $n\ne m$.

A generic element of ${⨁}_n{\mathcal{H}}_n$ is therefore a superposition across all sectors, not a single vector in one ${\mathcal{H}}_n$. The "particle number" $\hat{N}={\sum }_{n}n{\hat{P}}_{{\mathcal{H}}_n}$ has integer eigenvalues, but generic states (coherent states, the interacting vacuum, ...) are not eigenstates of $\hat{N}$.

0.0.8 Putting It Together: Fock Space

With this machinery, the Fock space definitions used below are:

$${\mathcal{F}}_{\text{boson}}({\mathcal{H}}_1)=\underset{n=0}{\overset{\infty }{⨁}}{\mathrm{Sym}}^n({\mathcal{H}}_1),$$ $${\mathcal{F}}_{\text{fermion}}({\mathcal{H}}_1)=\underset{n=0}{\overset{\infty }{⨁}}{\Lambda }^n({\mathcal{H}}_1).$$

Each Fock space is a Hilbert direct sum of $n$-particle sectors ${\mathcal{H}}_n$, where each ${\mathcal{H}}_n$ is the (anti)symmetric tensor power of the single-particle space ${\mathcal{H}}_1$. The vacuum $|0⟩\in \mathbb{C}={\mathcal{H}}_0$ sits at the bottom; the single-particle space ${\mathcal{H}}_1$ sits at $n=1$ and is the same as before second quantization (see §0.2 below).

0. Before vs. After Second Quantization

It is worth pausing to ask: what was the Hilbert space before second quantization, and how does it relate to Fock space? Answering this carefully clears up a recurring confusion: are the "states" of the pre-QFT theory the same as the states of the QFT?

0.1 Two Quantizations, Not One

The historical route to QFT involves two distinct constructions, both called "quantization":

The terminology is regrettable — there is only one Hilbert-space construction (first quantization); the second step is really field quantization — but the historical names have stuck.

Note on what "stage 0" means. Several different objects could legitimately be called "the classical field":

Object	Components	Source of the count
Classical relativistic particle (worldline $x^{\mu }(\tau )$)	4 spacetime coords, 0 internal components	Spacetime dimension
Classical Dirac field ${\psi }_{\text{cl}}(x)$	4 spinor components per spacetime point	Clifford algebra (see §0.8.1) — not the spacetime dimension
Classical Maxwell field $A_{\mu }(x)$	4 components per spacetime point	Spacetime dimension (genuinely; $\mu =0,1,2,3$)

The table above uses the classical Dirac field as "stage 0" — the modern field-theory starting point. Note that this is not what Dirac himself started from historically: he began with a relativistic particle (no field), found his equation by demanding a first-order quantization (postulate H1), and only later did anyone write down the corresponding classical Lagrangian. So the modern chain "classical field → first quantization → second quantization" is a retrospective tidying-up; the original sequence was "classical particle → relativistic wavefunction → field operator". Either ordering arrives at the same QFT.

Note also that the two "4"s in the Dirac and Maxwell rows above are different things: spinor index vs. spacetime index. They coincide only in 4D spacetime — see the caveat in §0.8.1.

0.2 Pre-Quantization Setup (Single-Particle Relativistic QM)

After first quantization but before second quantization:

• Hilbert space: ${\mathcal{H}}_1\cong L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^4$ (Dirac case).
• State: a single vector $|\psi ⟩\in {\mathcal{H}}_1$, i.e. a wavefunction $\psi (\mathbf{x},t)\in {\mathbb{C}}^4$.
• Particle number: fixed at $N=1$ by construction.
• Operators: $\hat{x}$, $\hat{p}$, $\hat{S}$, etc. — they act on a single wavefunction; they cannot create or destroy particles.
• Missing: no vacuum, no antiparticles (only awkward Dirac-sea heuristics), no scattering with changing particle number, no second particle.

For multi-particle relativistic QM (which is rarely written down because it does not really work as a stand-alone theory), one would put $N$ identical particles into

$${\mathcal{H}}_N={\mathrm{Sym}}^N({\mathcal{H}}_1)\text{or}{\Lambda }^N({\mathcal{H}}_1),$$

separately for each $N$, with no operators connecting different $N$ sectors.

0.3 Post-Quantization Setup (QFT)

After second quantization:

• Hilbert space: Fock space $\mathcal{F}={⨁}_{n=0}^{\infty }{\mathcal{H}}_n$ — the same ${\mathcal{H}}_n$ as direct summands, plus the new vacuum sector ${\mathcal{H}}_0$.
• State: a vector in $\mathcal{F}$, generically a superposition over different particle-number sectors.
• Particle number: no longer fixed. The number operator $\hat{N}$ has integer eigenvalues $0,1,2,\dots$, but generic states (coherent states, the interacting vacuum) are not eigenstates of $\hat{N}$.
• Operators: $b,b^{†},d,d^{†},a,a^{†}$ that connect different sectors; field operators $\hat{\psi }(x)$, ${\hat{A}}_{\mu }(x)$; observables $\hat{H},\hat{P},\hat{Q}$.
• Gained: vacuum, antiparticles as a derived concept (not the Dirac sea), variable particle number, locality of operators, automatic (anti)symmetrization.

0.4 Are the States "the Same"?

The one-particle states are essentially the same; everything else is genuinely new.

So every pre-quantization state has a faithful image in $\mathcal{F}$, but $\mathcal{F}$ contains many states with no pre-quantization counterpart.

0.5 The Mathematical Relationship

Concretely, Fock space is built from the single-particle space:

$${\mathcal{F}}_{\text{boson}}({\mathcal{H}}_1)=\mathbb{C}\oplus {\mathcal{H}}_1\oplus {\mathrm{Sym}}^2({\mathcal{H}}_1)\oplus {\mathrm{Sym}}^3({\mathcal{H}}_1)\oplus \cdots$$ $${\mathcal{F}}_{\text{fermion}}({\mathcal{H}}_1)=\mathbb{C}\oplus {\mathcal{H}}_1\oplus {\Lambda }^2({\mathcal{H}}_1)\oplus {\Lambda }^3({\mathcal{H}}_1)\oplus \cdots$$

Consequences:

• The single-particle space ${\mathcal{H}}_1$ is a closed subspace of $\mathcal{F}$ — no information is lost.
• The multi-particle sectors ${\mathcal{H}}_N$ are also subspaces — but pre-quantization theory needed a separate construction for each $N$, while Fock space gives them all at once.
• The truly new ingredients are (i) the vacuum ${\mathcal{H}}_0$, (ii) operators connecting different sectors, and (iii) coherent superpositions across sectors.

0.6 Why Bother?

Two reasons that are not obvious until you do it:

1. Operators that change particle number become local fields. The map $b_{\mathbf{x},s}^{†}|0⟩=$ "particle at $\mathbf{x}$" lets you write down operators like ${\hat{\psi }}^{†}(x)\hat{\psi }(x)$ — local densities, currents, energy-momentum tensors — that have no analog in pre-quantization theory because there is no vacuum to create from. This is what makes interactions and locality possible.
2. Antisymmetrization becomes automatic. In pre-quantization theory you postulate Slater determinants by hand and have to keep track of permutation signs. In second quantization the anticommutator $\{b_{\mathbf{x}}^{†},b_{\mathbf{y}}^{†}\}=0$ does this for you — Pauli exclusion is built into the algebra of operators.

0.7 A Useful Slogan

First quantization turns a classical particle into a quantum state. Second quantization turns a quantum state into a quantum operator.

Or, in terms of objects:

Classical:        position x(t)         field φ(x)            (numbers/functions)
                       ↓ first q.            ↓ first q.
1-particle QM:    operator x̂              wavefunction ψ(x)    (acts on / is in 𝓗_1)
                                              ↓ second q.
QFT:                                    field operator ψ̂(x)    (acts on 𝓕)

The right-hand column is the relevant one: a wavefunction (a state) is promoted to a field operator on a larger Fock space. The symbol $\psi (\mathbf{x},t)$ becomes $\hat{\psi }(\mathbf{x},t)$, but it now means an operator, not a state. Meanwhile the state of the system becomes a vector in Fock space, which is something genuinely new (see QFT/preliminaries.md § States vs. Fields).

0.8 The Pre-Quantization Hilbert Space ${\mathcal{H}}_1$ in Detail (Dirac Case)

The single-particle Dirac Hilbert space ${\mathcal{H}}_1$ looks deceptively simple — "square-integrable 4-component spinor wavefunctions" — but several things are non-obvious and worth spelling out, because they motivate why second quantization is needed in the first place.

0.8.1 The Naïve Definition

${\mathcal{H}}_1$ is an abstract complex separable Hilbert space carrying the spin-$\frac{1}{2}$, mass-$m$ representation of the (universal cover of the) Poincaré group. The most familiar concrete realization is the position representation

$${\mathcal{H}}_1\cong L^2({\mathbb{R}}^3,d^{3}x)\otimes {\mathbb{C}}^4,$$

in which a state vector $|\psi ⟩$ is represented by a 4-spinor wavefunction

$$\psi (\mathbf{x})=⟨\mathbf{x}|\psi ⟩=\left(\begin{array}{c}{\psi }_1(\mathbf{x})\\ {\psi }_2(\mathbf{x})\\ {\psi }_3(\mathbf{x})\\ {\psi }_4(\mathbf{x})\end{array}\right),\int d^{3}x|{\psi }_a(\mathbf{x}){|}^2<\infty .$$

Why 4 components? The number is forced, not chosen. Three equivalent derivations:

• Algebraic. Postulate H1 (first-order relativistic equation, see QED/historical.md §1.2) requires ${\gamma }^{\mu }$ satisfying the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }1$. The smallest faithful matrix representation of this algebra in 4D Minkowski space has dimension $2^{\lfloor 4/2\rfloor }=4$. (In $1+1$ D it would be $2$; larger sizes are reducible direct sums.) The general pattern — Bott periodicity, when Majorana/Weyl spinors exist — is collected in math/clifford-algebra.md. Hence $\psi$ must be a 4-component column.
• Group-theoretic. Under Lorentz transformations $\psi$ must carry a representation of $SL(2,\mathbb{C})$ (the universal cover of the Lorentz group). The smallest faithful spinor reps are the two 2-component Weyl spinors $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$. A massive particle's mass term $m\bar{\psi }\psi$ couples both chiralities, so a massive spin-$\frac{1}{2}$ field needs $(\frac{1}{2},0)\oplus (0,\frac{1}{2})$ — dimension $2+2=4$.
• Physical. The 4 components decompose as ${\mathbb{C}}^4\cong {\mathbb{C}}_{\text{particle/antiparticle}}^2\otimes {\mathbb{C}}_{\text{spin}}^2$: two for the spin-$\frac{1}{2}$ doublet, doubled by the antiparticle degree of freedom that relativity unavoidably introduces (§0.8.5–§0.8.7).

Non-relativistic spin-$\frac{1}{2}$ (Pauli theory) gets away with ${\mathbb{C}}^2$ because it has no antiparticle sector and uses Galilean rather than Lorentz boosts.

Caveat: the "4" in ${\mathbb{C}}^4$ is not the spacetime dimension. It is a coincidence that the spinor dimension equals the spacetime dimension in $d=4$. The spinor dimension is $2^{\lfloor d/2\rfloor }$ in $d$ spacetime dimensions:

Spacetime dim $d$	Spinor dimension $2^{\lfloor d/2\rfloor }$
$1+1=2$	$2$
$1+2=3$	$2$
$1+3=4$	$4$ ← our universe
$1+4=5$	$4$
$1+5=6$	$8$
$1+9=10$	$32$ (relevant in superstring theory)

In 6D spinors would be 8-component, in 10D 32-component. The "4 of ${\gamma }^{\mu }$" (counting $\mu =0,1,2,3$) and the "4 of ${\mathbb{C}}^4$" (counting spinor components) are two different things that happen to coincide only in our 4D spacetime. The spatial dimension enters ${\mathcal{H}}_1$ separately, in the ${\mathbb{R}}^3$ inside $L^2({\mathbb{R}}^3)$.

Equivalent realizations of the same Hilbert space are obtained by unitary change of basis:

None of these is "the" Hilbert space — the Hilbert space is the abstract equivalence class; the rest are coordinate choices that diagonalize different operators (position, momentum, energy, ...).

There is an additional, independent choice for the ${\mathbb{C}}^4$ factor: a representation of the Clifford algebra (Dirac, Weyl, Majorana — see QED/historical.md §0.1 for the explicit matrix forms and math/clifford-algebra.md § 7 for the abstract classification). Different Clifford-algebra representations are related by a similarity transformation $\psi \to S\psi$, ${\gamma }^{\mu }\to S{\gamma }^{\mu }{S}^{-1}$, and again give the same abstract ${\mathcal{H}}_1$.

The position representation is used below by default because the Dirac equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi (x)=0$ is most familiar in that form. Time is not part of the Hilbert-space label; it parametrizes how a vector evolves under the Schrödinger-form Dirac equation.

0.8.2 The Inner Product

$$⟨\psi |\varphi ⟩=\int d^{3}x{\psi }^{†}(\mathbf{x})\varphi (\mathbf{x})=\int d^{3}x\sum _{a=1}^4{\psi }_a^{\ast }(\mathbf{x}){\varphi }_a(\mathbf{x}).$$

Note carefully: this uses ${\psi }^{†}\varphi$ (Hermitian conjugate), not the relativistic-looking $\bar{\psi }\varphi ={\psi }^{†}{\gamma }^0\varphi$. The latter is a Lorentz scalar but not positive-definite — it cannot be an inner product. The price of the former is that the inner product is not manifestly Lorentz-invariant: it singles out the time component of ${\gamma }^{\mu }$. This is one of the first signs that ${\mathcal{H}}_1$ does not sit comfortably with relativity.

The norm is $\Vert \psi {\Vert }^2=\int d^{3}x{\psi }^{†}\psi =\int d^{3}x{j}^0$, where $j^{\mu }=\bar{\psi }{\gamma }^{\mu }\psi$ is the Dirac probability current. Since $\bar{\psi }{\gamma }^0\psi ={\psi }^{†}\psi \ge 0$, the norm is positive-definite — this was the whole point of demanding a first-order equation (postulate H1 in QED/historical.md).

0.8.3 The Three Tensor-Product Factors

It helps to unpack $L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^4$ explicitly:

In the standard (Dirac) representation, the four components split as $\psi =\left(\begin{array}{c}\chi \\ \eta \end{array}\right)$, where $\chi$ ("upper") dominates for non-relativistic particle states and $\eta$ ("lower") dominates for antiparticles. In the Weyl (chiral) representation the same four components split into left- and right-handed Weyl spinors instead. Different representations of the Clifford algebra give different physical interpretations of the four components, but the abstract Hilbert space ${\mathcal{H}}_1$ is the same.

0.8.4 Position-Basis "Eigenstates" (Improper Vectors)

As in non-relativistic QM, position eigenstates $|\mathbf{x},a⟩$ ($a=1,\dots ,4$ a spinor index) are not normalizable elements of ${\mathcal{H}}_1$ — they are improper distributions:

$$⟨\mathbf{x},a|{\mathbf{x}}^{\prime },b⟩={\delta }_{ab}{\delta }^3(\mathbf{x}-{\mathbf{x}}^{\prime }),|\psi ⟩=\sum _a\int d^{3}x{\psi }_a(\mathbf{x})|\mathbf{x},a⟩.$$

They are the rigged-Hilbert-space generalized eigenvectors of the position operator $\hat{\mathbf{x}}=\mathbf{x}{1}_4$. They live in a larger space ${\Phi }^{\ast }\supset {\mathcal{H}}_1$, never in ${\mathcal{H}}_1$ itself.

0.8.5 Momentum Basis and the Free Hamiltonian

Going to momentum space by Fourier transform,

$$\tilde{\psi }(\mathbf{p})=\int \frac{d^{3}x}{(2\pi {)}^{3/2}}{e}^{-i\mathbf{p}\cdot \mathbf{x}}\psi (\mathbf{x}),$$

the Hilbert space is unitarily equivalent to $L^2({\mathbb{R}}^3,d^{3}p)\otimes {\mathbb{C}}^4$. The free Dirac Hamiltonian acts on each momentum mode as the $4\times 4$ matrix

$$h(\mathbf{p})=\mathbf{\alpha }\cdot \mathbf{p}+\beta m,\mathbf{\alpha }={\gamma }^0\mathbf{\gamma },\beta ={\gamma }^0,$$

whose eigenvalues are $E_{\pm }=\pm \sqrt{{\mathbf{p}}^2+m^2}$, each two-fold degenerate (for the two spin states). The corresponding eigenvectors are the c-number 4-spinors $u(p,s)$ (positive energy) and $v(p,s)$ (negative energy). This is the source of the negative-energy problem: half of the spectrum of ${\hat{H}}_{\text{Dirac}}$ on ${\mathcal{H}}_1$ is unbounded below.

0.8.6 Decomposition into Positive- and Negative-Energy Subspaces

Since $h(\mathbf{p}{)}^2=({\mathbf{p}}^2+m^2)1_4$, one can build orthogonal projectors

$${\Lambda }_{\pm }(\mathbf{p})=\frac{1}{2}\left(1\pm \frac{h(\mathbf{p})}{E_{\mathbf{p}}}\right),E_{\mathbf{p}}=+\sqrt{{\mathbf{p}}^2+m^2},$$

inducing an orthogonal decomposition

$${\mathcal{H}}_1={\mathcal{H}}_1^{(+)}\oplus {\mathcal{H}}_1^{(-)}.$$

Each summand is isomorphic to $L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^2$ (the ${\mathbb{C}}^2$ being the two-fold spin degeneracy). Two key facts:

• Each summand carries an irreducible unitary representation of the Poincaré group (mass $m$, spin $\frac{1}{2}$, positive or negative energy). ${\mathcal{H}}_1^{(+)}$ is the Wigner one-particle space — what Fock space ought to be built from.
• The decomposition is non-local in position space. The projectors ${\Lambda }_{\pm }$ become non-local integral kernels when transformed back to $\mathbf{x}$-space — a signature of the Newton–Wigner localization problem (§0.8.7).

After second quantization, ${\mathcal{H}}_1^{(+)}$ becomes the electron sector of Fock space, and ${\mathcal{H}}_1^{(-)}$ — with charge conjugation applied — becomes the positron sector, both with positive energy.

0.8.7 Why ${\mathcal{H}}_1$ Is Not a Satisfactory Relativistic Hilbert Space

Several pathologies show up if one tries to take ${\mathcal{H}}_1$ seriously as the state space of a relativistic quantum particle:

• Negative-energy spectrum. Already noted; the Hamiltonian is unbounded below.
• No relativistically invariant inner product. The natural Lorentz-scalar bilinear $\bar{\psi }\varphi$ is indefinite; the positive-definite ${\psi }^{†}\varphi$ singles out a frame.
• Newton–Wigner localization. There is no position operator on ${\mathcal{H}}_1$ whose eigenstates are Lorentz-covariantly localized. The naïve $\hat{\mathbf{x}}=\mathbf{x}{1}_4$ does not commute with the projection onto ${\mathcal{H}}_1^{(+)}$ — projecting a ${\delta }^3(\mathbf{x})$-localized state onto positive energies smears it out over a Compton wavelength $\hslash /(mc)$.
• Zitterbewegung. Solutions of the Dirac equation on ${\mathcal{H}}_1$ exhibit a rapid oscillation $\sim e^{2im{c}^{2}t/\hslash }$ between positive- and negative-energy components, with no classical analogue. It vanishes once one restricts to ${\mathcal{H}}_1^{(+)}$.
• Klein paradox. A barrier of height $>2m{c}^2$ "transmits" more current than is incident — pair production sneaking in through a single-particle interpretation.
• No multi-particle states. ${\mathcal{H}}_1$ has $N=1$ baked in; it cannot describe scattering processes that change particle number, which is exactly what relativistic kinematics permits.

All of these are symptoms of the same underlying issue: a relativistic theory cannot be a one-particle theory. Energy can be converted to particle number ($E=m{c}^2$), so any consistent theory must allow particle creation and annihilation. Second quantization is the resolution.

0.8.8 What Survives in QFT

Despite its problems as a relativistic state space, ${\mathcal{H}}_1$ — or really its positive-energy half ${\mathcal{H}}_1^{(+)}$ — has a clean and important role after second quantization:

• It is the one-particle sector of Fock space (the $n=1$ summand): ${\mathcal{H}}_1^{(+)}\subset \mathcal{F}$.

• It carries the Wigner irreducible representation for a mass-$m$, spin-$\frac{1}{2}$ particle.

• One-particle wavepackets

  $$|f,s⟩=\int \frac{d^{3}p}{(2\pi {)}^3}f(\mathbf{p})b_{\mathbf{p},s}^{†}|0⟩$$

  are vectors in ${\mathcal{H}}_1^{(+)}$, with $f(\mathbf{p})$ playing the role of the post-quantization "wavefunction" (see §3.1 below).

So the historical ${\mathcal{H}}_1$ does not disappear: its positive-energy half is reborn cleanly inside Fock space, and the negative-energy half is reinterpreted (via charge conjugation) as the antiparticle sector.

1. The State Space: Fock Space

After second quantization, the Hilbert space is no longer the single-particle space ${\mathcal{H}}_1$ (e.g. $L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^4$ for one Dirac particle), but the much larger Fock space

$$\boxed{\mathcal{F}=\underset{n=0}{\overset{\infty }{⨁}}{\mathcal{H}}_n,}$$

where:

• ${\mathcal{H}}_0=\mathbb{C}|0⟩$ — the one-dimensional vacuum sector.
• ${\mathcal{H}}_1$ — the one-particle Hilbert space (an irreducible Wigner representation of the Poincaré group, e.g. mass $m$, spin $\frac{1}{2}$).
• ${\mathcal{H}}_n$ — the $n$-particle space, (anti)symmetrized tensor product:

$${\mathcal{H}}_n=\{\begin{array}{ll}{\mathrm{Sym}}^n({\mathcal{H}}_1)& \text{bosons}\\ {\Lambda }^n({\mathcal{H}}_1)& \text{fermions}\end{array}$$

Concretely:

• A single fermion lives in ${\mathcal{H}}_1\cong L^2({\mathbb{R}}^3)\otimes {\mathbb{C}}^4$.
• A two-fermion state is an antisymmetrized two-particle wavefunction — a Slater determinant.
• An $n$-particle state has $n!$ permutations symmetrized/antisymmetrized.
• The vacuum $|0⟩$ is a distinguished one-dimensional sector — not "no wavefunction"; it is a normalized vector orthogonal to all ${\mathcal{H}}_{n\ge 1}$.

For a theory with several particle species, the full Fock space is a tensor product of one Fock space per species. For QED:

$${\mathcal{F}}_{\text{QED}}={\mathcal{F}}_{e^{-}}\otimes {\mathcal{F}}_{e^{+}}\otimes {\mathcal{F}}_{\gamma }.$$

2. The Operators

There are two distinct families.

2.1 Creation and Annihilation Operators

These are the building blocks of every other operator. For each mode (e.g. $(\mathbf{p},s)$ for an electron):

• $b_{\mathbf{p},s}^{†}$ — creates an electron in mode $(\mathbf{p},s)$. Raises sector: ${\mathcal{H}}_n\to {\mathcal{H}}_{n+1}$.
• $b_{\mathbf{p},s}$ — annihilates one. Lowers: ${\mathcal{H}}_n\to {\mathcal{H}}_{n-1}$.
• $d_{\mathbf{p},s}^{†}$, $d_{\mathbf{p},s}$ — analogous for positrons.
• $a_{\mathbf{k},\lambda }^{†}$, $a_{\mathbf{k},\lambda }$ — analogous for photons.

Algebra:

$$\{b_{\mathbf{p},s},b_{\mathbf{q},r}^{†}\}=(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q}){\delta }_{sr}\text{(fermion)},$$ $$[a_{\mathbf{k},\lambda },a_{{\mathbf{k}}^{\prime },{\lambda }^{\prime }}^{†}]=(2\pi {)}^3{\delta }^3(\mathbf{k}-{\mathbf{k}}^{\prime }){\delta }_{\lambda {\lambda }^{\prime }}\text{(boson)},$$

with all other (anti)commutators vanishing. These are densely-defined unbounded operators on $\mathcal{F}$.

2.2 Field Operators

The electron/positron field $\hat{\psi }(x)$ is not a state — it is an operator-valued distribution on Fock space:

$$\hat{\psi }(x)=\sum _s\int \frac{d^{3}p}{(2\pi {)}^3}\frac{1}{\sqrt{2{\omega }_{\mathbf{p}}}}(b_{\mathbf{p},s}u(p,s)e^{-ip\cdot x}+d_{\mathbf{p},s}^{†}v(p,s)e^{+ip\cdot x}).$$

So $\hat{\psi }(x)$ is a linear combination of creation and annihilation operators — at each spacetime point it is a 4-component (in spinor index) operator-valued distribution. The spinors $u(p,s)$ and $v(p,s)$ are c-number coefficients (numerical 4-vectors), not operators or states.

The photon field ${\hat{A}}_{\mu }(x)$ admits an analogous expansion in terms of $a$, $a^{†}$, and polarization vectors ${ϵ}_{\mu }(k,\lambda )$.

2.3 Observables Built from Fields

Energy, momentum, charge, and number are all expressible as integrals of bilinears in the fields (after normal-ordering to remove the zero-point divergences):

• Hamiltonian: $\hat{H}=\int \frac{d^{3}p}{(2\pi {)}^3}{\omega }_{\mathbf{p}}(b^{†}b+d^{†}d)+\int \frac{d^{3}k}{(2\pi {)}^3}|\mathbf{k}|a^{†}a$.
• Momentum: $\hat{\mathbf{P}}=\int \frac{d^{3}p}{(2\pi {)}^3}\mathbf{p}(b^{†}b+d^{†}d)+\cdots$.
• Electric charge: $\hat{Q}=e\int \frac{d^{3}p}{(2\pi {)}^3}(b^{†}b-d^{†}d)$.
• Number operators: ${\hat{N}}_e=\int \frac{d^{3}p}{(2\pi {)}^3}{b}^{†}b$, etc.

All act on $\mathcal{F}$.

3. The States

Once second quantization is in place, the "wavefunction" of pre-QFT QM is gone — its role is taken over by state vectors in Fock space, of which there are several useful kinds.

3.1 Single-Particle States Are Not Wavefunctions

Single-particle states $|p,s⟩$ are not the wavefunctions $\psi (x)$ of pre-QFT. They are vectors in the one-particle sector ${\mathcal{H}}_1\subset \mathcal{F}$. The connection to Dirac wavefunctions is through matrix elements of the field operator:

$$⟨0|\hat{\psi }(x)|p,s⟩=u(p,s)e^{-ip\cdot x}.$$

The right-hand side is a Dirac wavefunction — but it is the matrix element of an operator between two specific states, not the state itself.

The proper "single-particle wavefunctions" in the QFT framework are wavepackets:

$$|f,s⟩=\int \frac{d^{3}p}{(2\pi {)}^3}f(\mathbf{p})b_{\mathbf{p},s}^{†}|0⟩\in {\mathcal{H}}_1,$$

with $f(\mathbf{p})$ playing the role of the momentum-space wavefunction. Its position-space transform $f(\mathbf{x})$ is, in many ways, the closest analogue of the pre-QFT $\psi (\mathbf{x},t)$.

3.2 The Interacting Vacuum

The interacting (physical) vacuum $|\Omega ⟩$ is not the same as the free Fock vacuum $|0⟩$. Loop diagrams continually create and annihilate virtual particles, so $|\Omega ⟩$ has nonzero overlap with all ${\mathcal{H}}_n$ for the free Fock decomposition.

In fact, by Haag's theorem (see QFT/remarks.md), $|\Omega ⟩$ lives in a Hilbert space unitarily inequivalent to $\mathcal{F}$ — a deep structural issue that textbook QFT typically glosses over by working with formal power series and renormalization, never with $|\Omega ⟩$ explicitly.

In perturbation theory, $|\Omega ⟩$ is reached from $|0⟩$ by adiabatic switching of the interaction (Gell-Mann–Low), and most computations only ever require the correlator $⟨\Omega |T\hat{\varphi }(x_1)\cdots \hat{\varphi }(x_n)|\Omega ⟩$, not the state itself.

3.3 Density Operators (Mixed States)

Density operators $\hat{\rho }$ on $\mathcal{F}$ describe statistical mixtures: thermal QED ($\hat{\rho }=e^{-\beta \hat{H}}/Z$), decoherence, and partial-trace reduced states for entanglement studies. They are not vectors but positive-semidefinite, trace-class operators on $\mathcal{F}$ with $\mathrm{Tr}\hat{\rho }=1$.

Expectation values are then $⟨\hat{A}⟩=\mathrm{Tr}(\hat{\rho }\hat{A})$, generalizing the pure-state formula $⟨\psi |\hat{A}|\psi ⟩$.

4. What's "Wavefunction-Like" After Second Quantization

If you really want to recover something wavefunction-shaped from the Fock-space machinery, there are three places it shows up:

1. Single-particle wavepacket coefficient $f(\mathbf{p})$ or its Fourier transform $f(\mathbf{x})$ — same role as the QM wavefunction, but only for a one-particle sector.
2. $N$-particle wavefunction $\Phi ({\mathbf{x}}_1,\dots ,{\mathbf{x}}_N)=⟨0|\hat{\psi }({\mathbf{x}}_1)\cdots \hat{\psi }({\mathbf{x}}_N)|\Psi ⟩$ — recoverable in principle for a fixed-$N$ sector, but rarely useful in practice (relativistic kinematics, no fixed particle number under interactions).
3. Mode-function coefficients $u(p,s)$, $v(p,s)$, ${ϵ}^{\mu }(k,\lambda )$ — the c-number coefficients in the field expansion. Often called "wavefunctions" loosely, but they are Lorentz-covariant coefficient objects, not states.

5. Inventory at a Glance

┌──────────────────────────────────────────────────────┐
│  Fock space  𝓕 = ⨁ₙ 𝓗ₙ                              │
│                                                      │
│   States  (vectors)            Operators             │
│   ──────────────────           ──────────            │
│   |0⟩                          b, b†, d, d†, a, a†   │
│   |p,s⟩ = b†|0⟩                ψ̂(x), Â_μ(x)          │
│   |p₁,s₁;p₂,s₂⟩                Ĥ, P̂, Q̂, N̂            │
│   wavepackets                                        │
│   coherent states                                    │
│   bound states                 (built from a/a†)     │
│   density matrices ρ̂                                 │
│                                                      │
│   c-numbers (not on 𝓕)                               │
│   ─────────────────────                              │
│   u(p,s), v(p,s), ε^μ(k,λ)  — mode coefficients      │
│   ⟨0|ψ̂(x)|p,s⟩ = u(p,s)e^{-ip·x}  — matrix element   │
└──────────────────────────────────────────────────────┘

The two most common confusions:

• The field operator $\hat{\psi }(x)$ is not a state and not a wavefunction. It is an operator-valued distribution on $\mathcal{F}$.
• The mode-function spinors $u(p,s)$, $v(p,s)$ are c-numbers, not states. They appear as numerical coefficients multiplying $b$, $d^{†}$ in the field expansion, and as matrix elements $⟨0|\hat{\psi }(x)|p,s⟩$.